\[ \int \frac {(d+e x)^{5/2}}{\sqrt {c d^2-c e^2 x^2}} \, dx \]
Optimal antiderivative \[ -\frac {2 \left (e x +d \right )^{\frac {3}{2}} \sqrt {-c \,e^{2} x^{2}+c \,d^{2}}}{5 c e}-\frac {64 d^{2} \sqrt {-c \,e^{2} x^{2}+c \,d^{2}}}{15 c e \sqrt {e x +d}}-\frac {16 d \sqrt {e x +d}\, \sqrt {-c \,e^{2} x^{2}+c \,d^{2}}}{15 c e} \]
command
integrate((e*x+d)^(5/2)/(-c*e^2*x^2+c*d^2)^(1/2),x, algorithm="giac")
Giac 1.9.0-11 via sagemath 9.6 output
\[ \frac {2}{15} \, {\left (\frac {32 \, \sqrt {2} \sqrt {c d} d^{2}}{c} - \frac {60 \, \sqrt {-{\left (x e + d\right )} c + 2 \, c d} d^{2}}{c} + \frac {20 \, {\left (-{\left (x e + d\right )} c + 2 \, c d\right )}^{\frac {3}{2}} c d - 3 \, {\left ({\left (x e + d\right )} c - 2 \, c d\right )}^{2} \sqrt {-{\left (x e + d\right )} c + 2 \, c d}}{c^{3}}\right )} e^{\left (-1\right )} \]
Giac 1.7.0 via sagemath 9.3 output
\[ \int \frac {{\left (e x + d\right )}^{\frac {5}{2}}}{\sqrt {-c e^{2} x^{2} + c d^{2}}}\,{d x} \]________________________________________________________________________________________